Let $A=\mathbb{Z}$ and let $a~R~b$ mean that $a \mid b$. Is $R$ an equivalence relation on $A$?
I need to finish this exercise but I am not really sure how, I have to finish a list of exercises like that but I can't seem to get the main idea of this exercise. Can anyone help me so I can finish the others?
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To show that a relation is an equivalence relation, check that it is symmetric, transitive, and reflexive.
To prove that it is not, you just have to show that one of the condition fails.
For any integers, it divides itself since $x=x(1)$, so it is reflexive.
If $a|b$, can we say that $b|a$ for sure? $2|4$ but does $4|2$?
It is transitive, try to prove it.
To show that a relation is an equivalence relation, you just need to show that it is symmetric, transitive, and reflexive. Trivially "is divisible by" is reflexive. Is it symmetric and transitive?